Tunable, nonlinear acoustic metamaterials due to subwavelength structural instabilities
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This dissertation studies the fundamental behavior associated with a class of nonlinear acoustic metamaterials that derive their material properties from a random distribution of non-interacting hyperelastic inclusions with designed mechanical instabilities embedded in a nearly incompressible viscoelastic matrix material. A metamaterial is an effective element whose behavior originates due to the subwavelength structure and not the inherent mechanical properties of the constituents and can attain behavior, such as negative stiffness, that is unattainable with conventional materials. Often metamaterials utilize resonance phenomena and periodicity of unit cells to achieve the desired response; however, the present work focuses on negative stiffness induced via nonlinear structural elements with engineered instabilities. These instabilities induce a local stiffness that varies as a function of an applied pre-strain. Since acoustic phenomena of interest, such as harmonic generation or energy dissipation, are often on the macroscale, homogenization methods to define the macroscopic heterogeneous medium are necessary. The intent of this work is thus twofold. First, modeling techniques for the quasi-static and dynamic response of the micro- and macroscale are required. A nonlinear dynamic model is first developed to study the dispersive and frequency-dependent properties of the heterogeneous medium. A quasi-static approximation is obtained via the low-frequency limit of such a dynamic model. Then, a coupled multiscale models is obtained to study the subresonant dynamics and to predict energy dissipation due to the inclusion oscillations and nonlinearity via harmonic generation. In contrast, a purely quasi-static model to obtain effective material properties of the heterogeneous medium is obtained via an augmented Hashin-Shtrikman method that accounts for material and geometric nonlinearity up to cubic order. The models of interest are explored for one example inclusion design for the class of acoustic metamaterials of interest. The resulting material response on both the micro- and macroscales are obtained as a function of deformation, which offers the ability to tune and tailor the macroscopic response to achieve a desired result.